Lesson plan

Logarithmically concave function

In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality : f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta} for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, : \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y) for all and . Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality : f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta} for all and . For non-negative discrete functions , it is log-concave if : f(k)^2 \geq f(k+1)f(k-1)

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Article summary

In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality : f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta} for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, : \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y) for all and . Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality : f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta} for all and . For non-negative discrete functions , it is log-concave if : f(k)^2 \geq f(k+1)f(k-1)

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